Question

10 trigonometric ratios and functions exam
write the equation of the sinusoidal function shown.
a) y = sin x - 1
b) y = 2 sin x - 1
c) y = 2 cos x - 1
d) y = cos x - 1

Explanation:

Step1: Determine the amplitude

The amplitude $A$ of a sinusoidal function $y = A\sin(x)+k$ or $y=A\cos(x)+k$ is half of the vertical distance between the maximum and minimum values. The maximum value of the given graph is $1$ and the minimum is $- 3$. So, $A=\frac{1 - (-3)}{2}=\frac{4}{2}=2$.

Step2: Determine the vertical - shift

The vertical - shift $k$ is the mid - value between the maximum and minimum values. $k=\frac{1+( - 3)}{2}=\frac{-2}{2}=-1$.

Step3: Determine the type of function

The graph passes through the point $(0, - 1)$. For $y = A\sin(x)+k$, when $x = 0$, $y=k$ (since $\sin(0)=0$). For $y = A\cos(x)+k$, when $x = 0$, $y=A + k$. Since when $x = 0$, $y=-1$ and $A = 2,k=-1$, if we consider $y=A\sin(x)+k$, substituting $x = 0$ gives $y=-1$ (because $\sin(0)=0$).

Answer:

B. $y = 2\sin x-1$